The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy

Transcription

1 The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy Antony Jameson Stanford University Aerospace Computing Laboratory Report ACL 007 January 007

2 Abstract This work revisits an idea that dates back to the early days of scientific computing, the energy method for stability analysis. It is shown that if the scalar nonlinear conservation law u t + x f(u) = 0 is approximated by the semi-discrete conservative scheme du j dt + ( f x j+ f j ) = 0 then the energy of the discrete solution evolves at exactly the same rate as the energy of the true solution, provided that the numerical flux is evaluated by the formula f j+ = 0 f(û)dθ where û(θ) = u j + θ(u j+ u j ). With careful treatment of the boundary conditions, this provides a path to the construction of non-dissipative stable discretizations of the governing equations. If shock waves appear in the solution, the discretization must be augmented by appropriate shock operators to account for the dissipation of energy by the shock waves. These results are extended to systems of conservation laws, including the equations of incompressible flow, and gas dynamics. In the case of viscous flow, it is also shown that shock waves can be fully resolved by non-dissipative discretizations of this type with a fine enough mesh, such that the cell Reynolds number.

3 Introduction Throughout the history of the development of discrete methods for linear and nonlinear conservation laws, and in particular computational fluid dynamics, there has been an ongoing struggle to find schemes which both assure physically correct non-oscillatory discrete solutions, and also minimize the discretization errors. Standard shock capturing schemes are formulated to satisfy total variation diminishing (TVD) or local extremum diminishing (LED) properties through the addition of sufficient artificial diffusion. On the other hand it seems that if the conservation law has an accompanying energy estimate, it should be possible to construct discrete schemes which satisfy the same estimate, and must therefore be stable without the need to introduce artificial diffusion, at least as long as the solution remains smooth. The use of energy estimates to establish the stability of discrete approximations to initial value problems has a long history. The energy method is discussed in the classical book by Morton and Richtmyer [], and it has been emphasized by the Uppsala school under the leadership of Kreiss and Gustafsson. Consider a well posed intitial value problem of the form du dt = Lu (.) where u is a state vector, and L is a linear differential operator in space with approximate boundary conditions. Then forming the inner product with u, ( u, du ) dt If L is skew self-adjoint, L = L, and the right hand side is Then the energy (u, u) cannot increase. = d (u, u) = (u, Lu) (.) dt (u, Lu) + (u, L u) = 0 If (.) is approximated in semi-discrete form on a mesh as dv dt = Av (.3) where v is the vector of the solution values of the mesh points, the corresponding energy 3

4 balance is and stability is established if d dt (vt v) = v T Av (.4) v T Av 0 (.5) A powerful approach to the formulation of discretizations with this property is to construct A in a manner that allows summation by parts (SBP) of v T Av, annihilating all interior contributions, and leaving only boundary terms. Then one seeks boundary operators such that (.5) holds. In particular suppose that A is split as A = D + B where D is an interior operator and B is a boundary operator. Then if D is skew-symmetric, D T = D, the contribution v T Dv vanishes leaving only the boundary terms. Skew-symmetric and SBP operators of both second and higher order have been devised for a variety of problems. The benefits of kinetic energy preserving schemes for the treatment of incompressible viscous flow has also been emphasized by Moin and Stanford s Center for Turbulence Research. Honein and Moin have also examined skew-symmetric schemes for compressible flow [] SBP operators are typically constructed by splitting the equations into a part in conservation form and a part in quasilinear form. For example, the inviscid Burgers equation is written as u t + ( ) u 3 x + 3 u u x = 0 Then the use of second order central difference operators for both parts at every interior point, and one sided operators at each boundary yields an SBP operator. In nonlinear problems for which the solution may develop shock waves it is generally beneficial to preserve conservation form in the discretization. According to the theorem of Lax and Wendroff [3], this will assure that the discrete solution satisfies the correct shock jump conditions, provided that it converges in the limit as the mesh interval is reduced to zero. In any case it is highly desirable to maintain global conservation properties of the true solution in the discrete solution. Small errors in the global conservation of mass, for example, can lead to large errors in the solution of flows in ducts. This paper presents a general procedure for constructing semi-discrete schemes for con- 4

5 servation laws in conservation form such that the discrete solution also exactly satisfies a discrete global conservation law for a generalized energy or entropy function. To take advantage of this approach it is first necessary to identify an appropriate energy principle for the governing equations. If shock waves appear in the solution the energy principle will need to be modified to allow for their effect on the energy or entropy balance. Moreover, in the light of Godunov s theorem that monotonically varying discrete shocks can only be obtained by locally first-order accurate schemes, the basic discretization scheme will need to be augmented by appropriate shock operators. The construction of shock operators for the inviscid Burgers equation and for the gas dynamics equations has been discussed by Gustafsson and Olsson [4]. Shock capturing schemes for gas dynamics have been widely studied [Godunov, Boris, Van Leer, Roe, Harten, Liou, Jameson [5, 6, 7, 8, 9, 0,, ]. In general they add artificial diffusion either explicitly or implicitly through the use of upwind operators. The aim of the present work is to devise stable schemes which would require the introduction of artificial diffusion only on the neighborhood of discontinuities and nowhere else. The next section discusses the Burgers equation and the energy principle. It is shown how to construct a semi-discrete scheme in conservation form, together with boundary operators, such that the discrete energy is bounded from above by the energy of the true solution. When shock waves appear in the solution the energy balance has to be modified to account for the dissipation of energy by the shock waves. It is then shown how to construct a shock operator which enables the discrete solution to track the energy evolution of the true solution very accurately, as verified by numerical experiments. Section 3 presents a procedure for constructing semi-discrete approximations to general scalar conservation laws in conservation form such that a discret energy principle is satisfied exactly. The conditions for the construction of the numerical flux are given in Theorem 3.. Section 4 extends the method to the treatment of systems of conservation laws. The conditions for the construction of the numerical flux such that a generalized entropy balance is satisfied exactly are given in Theorem 4.3. The method requires that the system can be symmetrized by a transformation of variables, as described in the work of Godunov, Mock and Harten [3, 4, 5]. Section 5 applies the foregoing method to the treatment of three dimensional incompressible flow. Theorem 5. states the discrete energy principle that is exactly satisfied by semi-discrete finite volume schemes with polyhedral control volumes, as has also been veri- 5

6 fied numerically. Finally the treatment of the gas dynamics equations is discussed in Section 6. It is shown that a semi-discrete finite volume scheme can be constructed to satisfy exactly a discrete entropy balance as long as the numerical flux is constructed in the proper manner, as stated in Theorem 6.. The formula, however, for the numerical flux requires the introduction of entropy variables, and is expressed as an integral that does not have a closed form. It is suggested that it should be evaluated by Gauss Lobatto integration. 6

7 Burgers equation The Burgers equation is the simplest example of a nonlinear equation which supports wave motion in opposite directions and the formation of shock awaves, and consequently it provides a very useful example for the analysis of the energy method. Expressed in conservation form, the inviscid Burgers equation is where and the wave speed is u t + f(u) = 0, a x b, (.) x f(u) = u (.) a(u) = f u = u (.3) Boundary conditions specifying the value of u at the left or right boundaries should be imposed if the direction of u is towards the interior at the boundary. Smooth solutions of (.) remain constant along characteristics x ut = ξ If a faster moving wave over-runs a slower moving wave this would indicate a multi-valued solution. Instead a proper weak solution is obtained by assuming the formation of a shock wave across which there is a discontinuous transition between left and right state u L and u R. In order to satisfy the conservation law (.) in integral form, u L and u R must satisfy the jump condition f(u R ) f(u L ) = s(u R u L ) (.4) where s is the shock speed. For the Burgers equation this gives a shock speed s = (u R + u L ) (.5) Provided that the solution remains smooth, (.) can be multiplied by u k and rearranged to give an infinite set of invariants of the form t ( ) u k k + ( ) u k+ = 0 x k + 7

8 Here we focus on the first of these t ( ) u + x ( ) u 3 = 0 (.6) 3 This may be integrated over x from a to b to determine the rate of change of the energy in terms of the boundary fluxes as E = b a u dx (.7) de dt = u3 a 3 u3 b 3 (.8) This equation fails in the presence of shock waves, as can easily be seen by considering the initial data u = x in the interval [, ]. Then a wave moves inwards from each boundary at unit speed toward the center until a stationary shock wave is formed at t =, after which the energy remains constant. Thus E(t) = + t, 0 t 3 3, t > In order to correct (.8) in the presence of a shock wave with left and right states u L and u R, equation (.6) should be integrated separately on each side of the shock. If the shock is moving at a speed s there is an additional contribution to de dt ( ) u s L u R Substituting equation (.5) for the shock speed which can be simplified to in the amount de dt = u3 a 3 u3 L 3 + u3 R 3 u3 b 3 + ( ) u (u L + u R ) L u R de dt = u3 a 3 u3 b 3 (u L u R ) 3 (.9) In the presence of multiple shocks, each will remove energy at the rate (u L u R ) 3. 8

9 As was already observed by Morton and Richtmyer [, page 4], a skew-symmetric difference operator consistent with (.) for smooth data can be constructed by splitting it between conservation and quasilinear form as u t + ( ) u 3 x + 3 u u x = 0 Suppose this is discretized on a uniform mesh x j = j x, j = 0,,... n. Central differencing of both spatial derivatives at interior points yields the semi-discrete scheme du j dt = ( u 6 x j+ uj ) + 6 x u j (u j+ u j ) = 0, j =, n (.0) The skew symmetric operator is completed by the use of one sided schemes at each boundary du 0 dt = 3 x (u u 0) + 3 x u 0(u u 0 ) du n = dt 3 x (u n u n ) + 3 x u n(u n u n ) (.) Rewriting the quasilinear term as 6 x (u j+u j u j u j ) equations (.0) and (.) can be expressed in the conservation form du j dt + ( f x j+ f j ) = 0, j =, n (.) where and f j+ = 6 du 0 dt + x ( du n dt + x ( u j+ + u j+ u j + u j) ( f f n f n (.3) ) f 0 = 0 ) = 0 (.4) where f 0 = u 0, f n = u n (.5) 9

10 Now let the discrete energy be represented by trapezoidal integration as E = x ( ) u 0 + u n n + x j= u j (.6) Multiplying equation (.) by u j and summing by parts n x j= n du j u j dt = j= u j (f j+ Hence, including the boundary points, we find that f j ) = f u 0 f n+ u n de dt = u3 0 3 u3 n 3 (.7) which is the exact discrete analog of the continuous energy evolution equation (.8). The formulation so far does not include the boundary conditions. Suppose that boundary data u = g 0 should be imposed at x 0 if the left boundary x 0 is an inflow boundary, and correspondingly u = g n should be imposed at x n if the right boundary x n is an inflow boundary. It is convenient to introduce the positive and negative wave speeds a + (u) = max(a(u), 0), a (u) = min(a(u), 0) Then we modify the equations at the boundary points by adding simultaneous approximation terms (SAT), so that instead of (.4) we solve du 0 dt + x (f f 0 ) + τ x a+ 0 (u 0 g 0 ) = 0 du n dt + x (f n f n ) τ x a 0 (u n g n ) = 0 (.8) where the parameter τ determines the amount of the penalty if the boundary condition is not satisfied exactly. The linear case has been analyzed by Mattsson [6]. here we wish to ensure stability in the nonlinear case. It is evident from (.7) that outflow boundaries (u 0 < 0 or u n > 0) promote energy decay. Thus we need only consider the effect of inflow boundary conditions. For this purpose suppose that d dt E true is the rate of change of energy that would result if the boundary conditions were exactly satisfied. Then we wish to choose τ so that de dt 0 is

11 bounded from above by d dt E true de dt d dt E true (.9) Consider the construction at the left boundary assuming it is an inflow boundary. Suppose that a + 0 is evaluated as (u 0 + g 0 ). Omitting for the moment the contribution of the right boundary we find that d dt (E E true) = u3 0 3 g3 0 3 τ 4 u 0(u 0 g 0) Suppose now that u 0 has the value αg 0. Then the rate of change of the energy is modified by the cubic expression F (α)g 3 0 where F (α) = α3 3 3 τ 4 α(α ) Here g 0 should be positive if it is truly an inflow boundary condition, so we require F (α) to be nonpositive in the range of α corresponding to inflow, α >. However, F (α) = 0 when α = and u 0 = g 0, so its sign will change at α = unless this is a double root. Here F (α) = α τ 4 (3α ) and the condition F () = 0 yields τ = (.0) Then F (α) = 6 (α ) (α + ) and is non positive whenever α >. A similar analysis at the right boundary confirms that condition (.0) is sufficient to assure the favorable energy comparison (.9) whenever either boundary is an inflow boundary. Numerical experiments have been conducted to verify the stability of the semi-discrete scheme (..3) with the boundary conditions (.8.0). Shu s total variation diminishing (TVD) scheme [7], was used for time integration. Writing the semi-discrete scheme in the form du dt + R(u) = 0 (.) where R(u) represents the discretized spatial derivative, this advances the solution during

12 one time step by the three stage scheme u () = u (0) t R(u (0) ) u () = 3 4 u(0) + 4 u() 4 t R(u() ) u (3) = 3 u(0) + 3 u() 3 t R(u() ) where u (0) and u (3) denote the solution of the beginning and the end of the time step. If (.) satisfies the TVD property with forward Euler time stepping, Shu s scheme is also TVD for time steps satisfying the CFL condition a t. This property has been designated x strongly stability preserving (SSP) by Gottlieb [8]. Figure. displays snap shots of the solution with initial data u = x in [, ] at times t = 0,.5,,.5 using a grid with 56 intervals. The true solution is a straight line connecting wave fronts moving inwards from both boundaries at unit speed until a shock is formed at t =. It can be seen that the discrete solution closely tracks the true solution prior to the formation of the shock. After the shock is formed the discrete solution develops strong oscillations in a zone expanding outward from the shock towards both boundaries. This is consistent with the fact that according to equation (.7), the discrete energy continues to grow at the rate t when t > as illustrated in Figure., and the energy must be absorbed 3 in the solution. It is evident that the scheme must be modified to preserve stability in the presence of shock waves. It is well known from shock capturing theory [, ], that oscillations in the neighborhood of shock waves are eleminated by schemes which are local extremum diminishing (LED) or total variation diminishing (TVD). A semi-discrete scheme is LED if it can be expressed in the form du i dt = j a ij (u j u i ) (.) where the coefficients a ij 0, and the stencil is compact, a ij 0 when i and j are not nearest neighbors. This property is satisfied by the upwind scheme in which the numerical

13 (a) At t = 0.0 (b) At t = 0.5 (c) At t =.0 (d) At t =.5 Figure.: Evolution of the solution of the Burgers equation 3

14 Figure.: Discrete energy growth flux (.3) is replaced by f j+ u j if a j+ > 0 = u j+ if a j+ < 0 (u j+ + u j) if a j+ = 0 (.3) where the numerical wave speed is evaluated as a j+ = (u j+ + u j ) (.4) Moreover, the upwind scheme (.3) admits a stationary numerical shock structure with a single interior point. The LED condition only needs to be satisfied in the neighborhoods of local extrema, which may be detected by a change of sign in the first differences u j+ = u j+ u j. A shock operator which meets these requirements can be constructed as follows. The numerical flux (.3) can be converted to the upwind flux (.3) by the addition of a diffusive term of 4

15 the form The required coefficient is d j+ = α j+ u j+. α j+ In order to detect an extremum introduce the function R(u, v) = u v u + v = 4 u j+ + u j (u j+ u j ) (.5) where q is an integer power. R(u, v) = whenever u and v have opposite signs. When u = v = 0, R(u, v) should be assigned the value zero. Now set so that s j+ = when u j+ 3 s j+ ( = R and u j u j+ 3 q, u j ) (.6) have opposite signs which will generally be the case if either u j+ or u j is an extremum. In a smooth region where u j+ 3 and u j are is an undivided difference. In not both zero, s j+ is of the order x q, since u j+ 3 u j order to avoid activating the switch at smooth extrema, and also to protect against division by zero, R(u, v) may be redefined as where ɛ is a tolerance []. Finally the diffusion term is modified to R(u, v) = u v max {( u + v ), ɛ} (.7) d j+ = max(s j+ 3, s j+, s j )α j+ u j+ Thus the coeffcient is reduced to a magnitude of order x q in smooth regions, while it has the value α j+ in the neighborhood of a shock. The value q = 8 has proved satisfactory in numerical experiments. Figure.3 shows the evolution of the discrete solution for the same case as Figure., with initial data u = x in [, ], using the shock operator defined by equations (.5-.7). A stationary shock with a single interior point is formed when t = as expected. Figure.4 5

16 (a) At t = 0.0 (b) At t = 0.5 (c) At t =.0 Figure.3: Evolution of the solution of the Burgers equation with a switch 6

17 Figure.4: Discrete energy growth with a limiter confirms that the discrete energy grows at the rate t until the shock forms and then remains 3 constant. The difference between the discrete energy and the true energy of the stationary solution is x because of the zero value in the middle of the discrete shock. Once the numerical shock structure is established the additional diffusive terms only contribute to du j dt at the three points s, s and s + comprising the shock, for which The only non-zero values of u j+ Also u s =, u s = 0, u s+ = u s are =, u s+ = a s =, α s = 3 a s+ =, α s+ = 3 7

18 Thus the additional contribution to de dt due to the shock is s= u s (α s+ u s+ α s u s ) = α s u s u s α s+ u s+ u s+ = 3 This exactly cancels the contribution of 3 change of the discrete energy is zero. from the boundaries, so that the total rate of In the case of the viscous Burgers equation with the viscosity coefficient ν u t + x ( ) u = ν u x (.8) the energy balance is modified by the viscous dissipation. Multiplying by u, and integrating the right hand side by parts with u x assumes the form = 0 at each boundary, the energy balance equation de dt = u3 a 3 u3 b 3 ν b a ( ) u dx (.9) x instead of (.8). Suppose that u is discretized by a central difference operator at interior x points with one sided formulas at the boundaries corresponding to u = 0, x x (u j+ u j + u j ), j =, n x (u u 0 ) at the left boundary, (.30) x (u n u n ) at the right boundary as proposed by Mattsson [6]. Then summing by parts with the convective flux evaluated by (.3) as before, the discrete energy balance is found to be de dt = u3 0 3 u3 n n 3 ν (u j+ u j ) (.3) j=0 This enables the possibility of fully resolving shock waves without the need to add any additional numerical diffusion via shock operators. The convective flux difference f j+ f j 8

19 (a) At t = 0.0 (b) At t = 0.5 (c) At t =.0 (d) At t =.5 Figure.5: Evolution of the solution of the viscous Burgers equation 9

20 Figure.6: Discrete energy growth for the viscous Burgers equation can be factored as 3 x (u j+ + u j + u j )(u j+ u j ) Accordingly the semi-discrete approximation to equation (.8) can written as where and du j dt = a j+ (u j+ u j ) + a j (u j u j ) (.3) a j+ a j = ν x u j+ + u j + u j 3 x = ν x + u j+ + u j + u j 3 x The semi-discrete approximation satisfies condition (.) for a local extremum diminishing scheme if a j+ 0 and a j 0. This establishes theorem.: Theorem. The semi-discrete approximation (.) using the numerical flux (.3) and the central difference operator (.30) for u x is local extremum diminishing if the cell Reynolds 0

21 number satisfies the condition where the local speed is evaluated as ū x ν (.33) ū = 3 u j+ + u j + u j (.34) It has been confirmed by numerical experiments that shock waves are indeed fully resolved with no oscillation if the cell Reynolds number satisfies condition (.33). Figure.5 shows the evolution of the discrete viscous Burgers equation using this scheme for the same initial data as before, u = x in [, ]. The Reynolds number ul based on the boundary ν velocity u = ± and the length of the interval was 048, and the solution was calculated on a uniform mesh with 04 intervals, so that the cell Reynolds number condition (.33) was satisfied in the entire domain. It can be seen that a stationary shock wave is formed at the time t =, and it is finally resolved with three interior points. Correspondingly the energy becomes constant after the shock wave is formed, as can be seen in Figure.6.

22 3 The one dimensional scalar conservation law Consider the scalar conservation law u t + f(u) = 0 (3.) x u(x, 0) = u 0 (x), u specified at inflow boundaries. Correspondingly, smooth solutions of (3.) also satisfsy where since multiplying (3.) by u yields Defining the energy as t ( ) u + F (u) = 0 (3.) x F u = uf u u u t + uf u u x = 0 E = b a u dx it follows from (3.) that smooth solutions of (3.) satisfy the energy equation Introducing the function G(u) such that de dt = F (u a) F (u b ) (3.3) G u = f

23 and multiplying (3.) by u we obtain u u t + u f x = t Thus F and G can be identified as ( ) u + u (uf) f x x = ( ) u + t x (uf) G u u x = ( ) u + (uf G) t x = 0 F = uf G, G = uf F (3.4) For the inviscid Burgers equation F = u3 3, G = u3 6 Suppose now that (3.) is discretized on a uniform grid over the range j = 0, n. Consider a semi-discrete conservative scheme of the form du j dt + x (f j+ f j ) = 0 (3.5) at every interior point, where the numerical flux f j+ is a function of u i over a range bracketing u j such that f j+ = f(u) whenever u is substituted for the u i, thus satisfying Lax s consistency condition. Multiplying (3.) by u j and summing by parts over the interior points we obtain n du j x u j dt j= n = j= u j (f j+ f j ) = u f 3 u f 5 +u f + u f 3... u n f n 3 u n f n + u 3 f 5 = u f u n f n + n j=... + u n f n 3 f j+ (u j+ u j ) 3

24 Suppose now that where G uj+ f j+ = G uj+ (3.6) is the mean value of G u in the range from u j to u j+ such that G uj+ (u j+ u j ) = G(u j+ ) G(u j ) (3.7) Then, denoting G(u j ) by G j, n x j= u j du j dt n = u f u n f n + (G j+ G j ) j= Now let (3.) be discretized at the end points as = u f u n f n G + G n du 0 dt + x (f f 0 ), du n dt + x (f n f n ) (3.8) where f 0 = f(u 0 ), f n = f(u n ) and define the discrete approximation to the energy as Then E = x ( ) u 0 + u n n + x j= u j de dt = u 0 f 0 u n f n G 0 + G n = F 0 F n (3.9) Thus the energy balance (3.3) is exactly recovered by the discrete scheme. Equations (3.6) and (3.7) are satisfied by evaluating the numerical flux as f j+ = 0 f(û(θ))dθ (3.0) where û(θ) = u j + θ(u j+ u j ) (3.) 4

25 since then G j+ G j = = 0 0 G u (û(θ))u θ dθ G u (û(θ)dθ(u j+ u j ) Thus we have established Theorem 3.: Theorem 3. If the scalar conservation law (3.) is approximated by the semi-discrete conservative scheme (3.5), it also satisfies the semi-discrete energy conservation law (3.8) if the numerical flux f j+ is evaluated by equations (3.0) and (3.). In the case of Burgers equation formulas (3.0) and (3.) yields the same numerical flux that was defined in Section f j+ = u j+ + u j+ u j + u j 6 (3.) In the case of a general polynomial flux f(u) = uq q (3.3) note that and G(u) = u 0 f(v)dv = uq+ q(q + ) G j+ G j = Thus the numerical flux should be evaluated as f j+ q(q + ) (uq+ j+ uq+ j ) = u j+ u j q(q + ) (uq j+ + uq j uq j ). = q + (uq j+ + uq j+ u j... + u q j ) (3.4) If (3.0) cannot be evaluated in closed form one may approximate it by numerical integration. The Lobatto quadrature rule which uses the end points and n interior points is suitable for this purpose, giving an exact result for polynomials of degree up to n 3. Taking n = 3 5

26 yields Simpson s rule f j+ = 6 (f(u j+) + 4f which is exact for the Burgers equation. ( ) (u j+ + u j ) + f(u j )) In order to enforce appropriate inflow and outflow boundary conditions we introduce simultaneous approximation terms (SAT). Denote the wave speed by a(u) = f u and let a + = (a + a ), a = (a a ) Then we replace (3.8) by du 0 dt + x (f f 0 ) + τ x a+ (u 0 g 0 ) = 0 du n dt + x (f n f ) τ x a (u n g n ) = 0 (3.5) where g 0 and g n denote the boundary values that should be imposed at inflow boundaries, and the magnitude of the penalty for not exactly satisfying the boundary conditions is determined by the parameter τ. In the case of the polynomial flux f(u) = uq q, q even, let a(u L, u R ) be the average numerical wave speed between the states u L and u R such that a(u L, u R )(u R u L ) = f R f L. Then a(u L, u R ) = q u q R uq L u R u L Now at the left boundary, for example, modify the equation by simultaneous approximation terms so that du 0 dt = x (f f 0 ) τ = x (f f 0 ) x a+ (u 0, g 0 )(u 0 g 0 ) τ k x (uq 0 g0) q 6

27 Then if u 0 = αg 0, the penalty term modifies the rate of change of the discrete energy by F (α)g q 0 where F (α) = αq+ q + τ q α(αq ) F () = 0, and F (α) will cross the axis at α = unless F () = 0. Here F (α) = α q τ k ((q + )αq ) and F () = 0 if τ =, giving F (α) = q ( αq ) Then F (α) > 0 in the interval < α <, and F (α) < 0 when α >. Consequently F (α) 0 in the entire range α >, yielding a favorable modification of the energy growth for all in flow conditions. 7

28 4 Discrete conservation of energy or entropy for a one dimensional system In this section it is shown how to extend the procedure described in Section 3 in order to discretely satisfy an additional invariant for a system of conservation laws. Consider the system u t + f(u) = 0 (4.) x where u now denotes the state vector, and f(u) is the flux vector. Suppose that h(u) is a convex function of u that can be regarded as an energy density. Multiplying (4.) by w T = h u we obtain u h u t = h t = wt f x = f T w x x (f T w) Suppose also that there exists a scalar function G(u) such that G w = G u u w = f T (4.) Then where the flux for h is h t = G w w x x (f T w) = F x (4.3) F = f T w G (4.4) The relation (4.) implies that f w = G ww and hence f w is symmetric. In the case of a system for which f u is not symmetric, such as the equations of gas dynamics, this precludes the choice h(u) = u T u The conditions under which a system of conservation laws can be written in symmetric hyperbolic form have been studied in papers by Godunov, Mock and Harten [3, 4, 5]. 8

29 Suppose that there exists a convex function h(u) such that h u f u = F u (4.5) for some scalar function F (u). Then Thus h(u) satisfies the conservation law h t = h u u t = h u uf u x = F u u x t h(u) + F (u) = 0 (4.6) x Here h(u) is a generalized entropy function and F (u) is the corresponding entropy flux. The main theorems are as follows: Theorem 4. Suppose (4.) can be symmetrized by a change of variables from u to w, so that it assumes the form w u w t + f w w x = 0 where u w and f w are symmetric and u w is positive definite. Then u = q w, u w = q ww for some convex function q(w), while f = G w, f w = G ww and (4.) has an entropy function h(u) = u T w q(w) (4.7) and entropy flux F (u) = f T w G(w) (4.8) Theorem 4. Suppose h(u) is an entropy function for (4.). Then w T = h u 9

30 symmetrizes (4.). The proofs are given by Harten [5]. Suppose now that (4.) is approximated in semi-discrete form on a uniform grid over the range j = 0, n as du j dt + x (f j+ f j ) = 0 (4.9) where the numerical flux f j+ is a function of u i over a range bracketing u j. Then we can construct a scheme which discretely satisfies the energy or entropy conservation law in the same manner as for the scalar case. Multiplying (4.9) by w T j = h uj and summing by parts over the interior points n wj T j= du j dt = n j= n u j h uj t = j= dh j dt n = w T f wn f T n + f T (w j+ j+ T wj T ) j= Now suppose that f T j+ = G wj+ (4.0) where G wj+ is a mean value of G w between w j and w j+ in the sense of Roe, such that G wj+ (w j+ w j ) = G j+ G j (4.) where G j denotes G(w j ). Then n x j= Now let (4.) be discretized at the end points as dh j dt = wt f wn f T n G + G n du 0 dt + x (f f 0 ) = 0, du n dt + x (f n f n ) = 0 where f 0 = f(u 0 ), f n = f(u n ) 30

31 Then we obtain the discrete conservation law x ( dh0 dt + dh ) n dt n + x j= dh j dt = w T 0 f n w T n f n G 0 + G n = F 0 F n (4.) where F is the entropy flux (4.4). G wj+ can be constructed to satisfy (4.) exactly in the form G wj+ = 0 G w (ŵ(θ)) dθ where ŵ(θ) = w j + θ(w j+ w j ) (4.3) since then G j+ G j = = 0 0 G w (ŵ(θ)) w θ dθ G w (ŵ(θ)) dθ(w j+ w j ) Thus we can state theorem 4.3: Theorem 4.3 The semi-discrete conservation law (4.9) satisfies the semi-discrete entropy conservation law (4.) if the numerical flux f j+ where ŵ(θ) is defined by (4.3). f j+ = 0 is constructed as f (ŵ(θ)) dθ (4.4) For some systems, such as the equations of gas dynamics, it may not be possible to express the integral (4.4) in closed form. Then one may rescale the interval of integration for θ to (-,) so that f j+ = f ( w(θ)) dθ 3

32 where w(θ) = (w j+ + w j ) + θ(w j+ w j ) and apply the n point Lobatto rule. In general neither boundary is necessarily purely inflow or outflow. Consequently, in order to impose proper boundary conditions it is essential to distinguish ingoing and outgoing waves at the boundaries. For this purpose we can split the Jacobian matrix A = f u into positive and negative parts A ±. Suppose that A is decomposed as A = RΛR where the columns of R are the right eigenvectors of A, and Λ is a diagonal matrix comprising the eigenvalues. Then A ± = RΛ ± R where Λ + and Λ contain the positive and negative eigenvalues respectively. Now the boundary conditions maybe imposed by adding simultaneous approximation terms (SAT) at the boundaries. Accordingly we set du 0 dt + x (f du n dt + x (f n f n f 0 ) + τ x A+ (u 0 g 0 ) = 0 ) τ x A (u n g n ) = 0 (4.5) where g 0 and g n define the exterior data and the parameter τ determines the magnitude of the penalty when the solution is not consistent with the incoming waves. Appropriate values of A 0 and A n may be obtained by taking them to be the mean valued Jacobian matrices in the sense of Roe [8] such that A 0 (u 0 g 0 ) = f(u 0 ) f(g 0 ) A n (u n g n ) = f(u n ) f(g n ) (4.6) If shock waves appear in the solution, the scheme needs to be modified by shock operators. A desirable property of a shock capturing scheme is that the numerical shock structure for a stationary shock should contain no more than one interior point []. This can be achieved by characteristic upwind schemes or the CUSP scheme []. The characteristic scheme can be constructed by adding matrix diffusion. Let f j denote f(u j ). Following Roe 3

33 [8], we introduce the mean Jacobian matrix A j+ such that A j+ (u j+ u j ) = f j+ f j Decomposing the Jacobian matrix in terms of its eigenvectors and eigenvalues as A j+ = RΛR as in the treatment above of the boundaries, define the absolute Jacobian matrix as A j+ = R Λ R where Λ is a diagonal matrix containing the absolute values of the eigenvalues. Then the upwind flux can be expressed as f Uj+ = (f j+ + f j ) A j+ (u j+ u j ) (4.7) Also let f Cj+ be the central flux defined by equation (4.4). Then we construct the flux throughout the domain as f j+ ( = S j+ ) f Cj+ + S j+ f Uj+ (4.8) where S j+ is a switching function with values in the range 0 S j+, of the order of a high power of x except in the neighborhood of a shock wave, where it should have a value of unity. The switching function can be constructed in a manner similar to the switch used for the Burgers equation, equations (.6) and (.7). The same formulas may be used to detect extrema in either the pressure or the entropy. Alternatively we can use these formulas to identify extrema in the the characteristic variables by applying them to v j+ = R u j+ j+ 33

34 5 Incompressible fluid flow The procedure for discretely satisfying the basic conservation laws plus an additional invariant, which has been proposed in Section 4, requires the systems to be expressed in symmetric form so that (4.) holds. In the case of incompressible flow the pressure is not directly related to the state vector via an equation of state, and must be determined indirectly from the continuity equation. Denoting the velocity components by v i, and the pressure and density by p and ρ, three dimensional inviscid incompressible flow is described by the continuity equation and three momentum equations ( vi ρ t + v j ) v i x j v i x i = 0 (5.) + p x i = 0, i =,, 3 (5.) In order to express the momentum equations in symmetric conservation form introduce the reduced pressure ˆp = p ρq (5.3) where q = v i. For convenience suppose that units are chosen such that ρ =. Using (5.), the momentum equations can be expressed as These can be written in state vector form as v i t + (v i v j ) + ( ) ˆp + v j = 0 (5.4) x j x i u t + f i (u) + P i = 0 (5.5) x i x i Here the state and flux vectors are u = v v v 3, f i q (u) = v i u + e i (5.6) where e i is the unit vector in the i direction and P i = e i ˆp (5.7) 34

35 Then and the Jacobian matrices are v i u = et i, u ( ) q = u T A i = f i u = v ii + ue T i + e i u T (5.8) The flux vectors f i are homogeneous functions of the state variables of degree two, with the consequence that A i u = f i (u) (5.9) Moreover where Multiplying (5.4) by v i f it (u) = Gi u G i (u) = u i q v i v i t + v i (v i v j ) + v i x j x i (5.0) = 3 ut f i (u) (5.) ( ) ˆp + v j = 0 (5.) Interchanging the indices in the second term, it may be grouped with the last term to give v j v j (v i v j ) + v i v j = x i x i x i (v i v j ) Thus (5.) reduces to ( ) v i + (v i vj + v iˆp) ˆp v i = 0 t x i x i Defining the energy flux as ( ) F i (u) = v i (ˆp + q ) = v i p + q = v i p t (5.3) where p t is the total pressure, we obtain the energy conservation law ( ) v i + F i t x i = ˆp v i x i (5.4) 35

36 where the right hand side vanishes provided that the continuity equation is satisfied. The total kinetic energy in a domain D with volume element dv is E = D v i dv Now (5.8) may be integrated to give de dt = n i F i ds + B ˆp v i x i dv (5.5) where B is the boundary of D, n i is the inward normal and ds is the area element. Suppose now that the domain is covered by a grid, and the equations are discretized in finite volume form with the discrete flow variables defined at the nodes, each of which is contained in a polyhedral control volume. In the case of either a hexahedral or a tetrahedral grid the control volumes may be taken as the dual cells connecting the centers of the primary cells. For example, one could construct the dual cells of a tetahedral mesh by dividing each primary cell about its median into subcells of equal volume which are assembled at the nodes to form the median dual cells. Examples of discretization schemes on the median dual mesh include the Airplane code of Jameson, Baker and Weatherill [9], and Stanford University s CDP code [0]. A representative two-dimensional grid is shown in Figure 5. The control volumes of the boundary nodes extend into the interior and are closed by faces on the boundary. Now the control volume of each interior node, say node o, consists of faces (not necessarily planar) with a directed face area S op for the face separating nodes o and p. The control volume of a boundary node o is closed by a vector face area S o which is the negative of the sum p S op of the face areas between o and each of its neighbors. The semi-discrete finite volume scheme has the form du o dt + ( ) Sop i f i vol op + Pop i o p (5.6) where the sum is over the faces of the control volume containing o, S i op is the projected area in the i direction of the face separating o and p, f i op and P i op are the convective and pressure fluxes between o and p, and the repeated superscript i denotes summation over the coordinate directions. At a boundary node there is an additional contribution S i of i o where f o = f(u o ) is the 36

37 Figure 5.: Convergent-divergent duct: = nodes of primary cells, = dual cells flux evaluated with the nodal values of the state vectors. The discrete energy is the sum E = over the nodes o, and its rate of change is vol o u T o u o (5.7) o de dt = o vol o u T o du o dt (5.8) du where vol o o is given by the sum (5.6) over the faces for that node. Each interior face dt appears twice in the resulting double sum because it is contained in the sums (5.6) for the two nodes it separates, while each exterior face appears once. The convective contribution of each interior face to de dt is thus ( ) u T p u T o S i op fop i where the flux f it op = Gi op u (5.9) 37

38 Provided f i op is constructed so that f it op (u p u o ) = G i p G i o (5.0) exactly, the convective contribution of each interior face is thus ( ) Sop i G i p G i o If one associates all terms containing G o with the node o, it receives the inner product of the sum of its face areas S op with G o, but this sum is zero. Thus the only contribution of the convective terms to de is a sum over the boundary nodes. However, the sum of the dt vector areas S op for a boundary node is the negative of its external area S o, so the convective contribution to de dt is boundary nodes S i o G i o There remains in the sum (5.8) the convective contribution boundary nodes S i o u T o f i o Thus the total convective contribution to de dt boundary nodes is ( ) So i u T fo i G i o (5.) where S o is the outward face area of node o. It remains to evaluate the contribution of the pressure to the discrete energy balance. For this purpose it is necessary to prove a discrete analog of the Gauss theorem D p v i dv = pv n ds p v i dv (5.) x i B D x i where v n is the normal velocity through the boundary. Across each face there is a contribution u T o p Si oppop i from the pressure. Suppose that the pressure flux is evaluated as P i op = ( P i o + P i p) (5.3) 38

39 The sum (5.8) then contains u T o Po i Sop i + o p o u T o Pp i Sop i p At every interior node o the fluxes u T p Sop i Po i generated by the neighbors can be associated with o, while u T o Po i p Si op = 0. Thus one can subtract this quantity to produce P it o (u p + u o ) Sop i while represents ˆp o vol ( u) o. A boundary node o receives the contributions P o it p p u p S i po = P it o from its neighbors while it retains the contributions giving the total P it o p (u p + u o ) S i op + P it o p u p Sop i P o it u o Sop i + Po it u o So i u o = P it o p p p Sop i + Po it u o So i (u p + u o )S i op P it o u o S i o + P it o u o S i o The first two terms represent ˆp o vol o ( u) o. The third term represents the surface integral on the right hand side of (5.). This establishes theorem 5. Theorem 5. The finite volume discretization of v i p x i o u T o p S i opp i op represented by where P i op is evaluated as the arithmatic average exactly satisfies the discrete analog of the Gauss Theorem (5.). Combining the boundary contribution from the pressure with the contribution (5.) of 39

40 the convective terms we obtain So(u i T fo i G i o u T Po) i = SoF i o i boundary nodes boundary nodes where Fo i is the discrete energy flux Fo i = vo i p to evaluated with the nodal values of v i and p t. This establishes theorem 5. Theorem 5. If the discrete flux vector f i op is constucted so that it satisfies (5.0), and the pressure flux is evaluated by (5.3), the rate of change of the discrete energy is exactly de dt = boundary nodes S i of i o + o vol ˆp o ( u) o where the second sum is over all the nodes and ( u) o is the consistent discretization of u at node o. Since f i (u) is a quadratic function of u, equation (5.0) is satisfied exactly by using Simpson s rule to evaluate f i op = = 6 f(u o + θ(u p u o ))dθ ( ( ) f(u o ) + 4f (u o + u p ) o ) + f(u p ) While the proof of theorem 5. is somewhat intricate, the author has numerically confirmed that it is satisfied to machine accuracy in test calculations for two different cases, flow past a circular cylinder, and flow through a convergent divergent duct. 40

41 6 Gas Dynamics In this section the theory developed in Section 4 is applied to the three dimensional gas dynamic equations. Because the Jacobians for the conservative variables are not symmetric, the formulation requires a transformation of variables which symmetrizes the equations. In conservation form the equations are u t + f i (u) = 0. (6.) x i Here the state and flux vectors are ρ ρv u = ρv ρv 3 ρe ρv i ρv i v + δ i p ρv i H, f i = ρv i v + δ i p ρv i v 3 + δ i3 p (6.) where ρ is the density, v i are the velocity components, and E and H are the specific energy and enthalpy. Also where γ is the ratio of specific heats. ( ) p = (γ )ρ E v i, H = E + p ρ (6.3) In the absence of shock waves the entropy ( ) p s = log ρ γ (6.4) is constant along streamlines, Harten shows in his paper [5] that s t + v s i = 0 x i h(s) = ρg(s), F i (s) = ρv i h(s) (6.5) 4

42 constitute a generalized entropy function and entropy fluxes provided that g(s) ġ(s) < γ (6.6) This equation ensures the convexity of h(u), which satisfies the entropy conservation law t h(u) + F i (u) = 0 (6.7) x i Moreover the conservation equations (6.) are symmetrized by the variables w T = h u (6.8) assuming the form w u w t + f i w = 0 (6.9) w x i where u w and f i w are symmetric, and u w > 0. Hughes, Franca and Mallet [] and Gerritsen and Olsson [] have proposed schemes using the entropy variables. Navier Stokes equations, choosing Reference [] presents a finite element formulation for the h(s) = ρs (6.0) in order to obtain a symmetric form for the viscous terms. Gerritsen and Olsson treat the Euler equations, and prefer the form ( ) h(s) = ρe s γ+α p γ+α = ρ ρ γ (6.) where α is a parameter to be chosen, because it leads to a homogeneous function u(w) such that u(θw) = θ β u(w) (6.) where and β = γ + α γ (6.3) u w w = βu (6.4) 4

43 The convexity condition (6.6) is satisfied if α > 0 or α < γ. The choice α = yields the comparatively simple form u 5 w 5 u w = p p u 3, u = p w p w 3 u 4 w 4 (6.5) u w where p = ) β e s γ+ = (γ ) (w w w 5 Multiplying (6.) by w T we obtain the entropy evolution equation in the form (6.6) The left hand side can also be expressed as u h u t = h(u) = wt f i (u) t x i w T u t = w wt u w t Gerritsen and Olsson split this as β w β + wt u w t + u wt β + t = = = ( u T w ) u + wt β + t t β + t (ut w) β + t (wt u w w). in view of the homogeneity relation (6.4). Thus h(u) can be regarded as an energy function for the system (6.). Gerritsen and Olsson also show that by splitting the spatial derivatives between the conservation and quasilinear forms as β f i (w) + f i w β + x i β + w x i and using central differencing for both at interior points, one obtains a skew-symmetric 43

44 operator which discretely satisfies the entropy conservation law (6.7). Since entropy will be generated by shock waves, artificial diffusion is still required for their capture. skew-symmetric form, which has been further investigated by Yee, Vinokur and Djomehri [3], gives up conservation form for the basic conservation laws (6.) in favor of discrete entropy conservation. However, the formulation of Section 4 actually allows discrete entropy conservation while retaining conservation form for the basic equations, as is shown in the following paragraphs. Consider a finite volume discretization of (6.) on a mesh similar to that described in Section 5, with u stored at the mesh nodes, each of which is contained in a control volume. The semi-discrete finite volume scheme at node o is p The du o dt + S vol opf i op i = 0 (6.7) o where the sum is over the faces of the control volume containing o, S i op is the projected area in the i direction of the face separating o and p, and f i op is the flux in the i direction between o and p. Then multiplying by vol o wo T entropy equation and summing over the nodes we obtain the discrete vol o wo T o du o dt = o vol o dh o dt = o w T o Sopf i op i (6.8) p As was shown in Section 5 the contribution of each interior face to this sum is (w T p w T o )S i opf i op where the flux can be expressed as f i op = G i w op (6.9) Provided that f i op(w) is constructed so that exactly, the contribution of each interior face is thus f it op (w p w o ) = G i p G i o (6.0) S i op(g i p G i o) 44

45 Associating all terms containing G i o with the node o, an interior node recovers a total contribution consisting of the inner product of the sum of the face areas S i op of its control volume with G i o, but the sum is zero. The sum of the face area S i op separating a boundary node from the neighbors is the negative of its external face area S i o, so the total contribution of the interior faces reduces to boundary nodes There remains in the sum (6.9) the contribution boundary nodes S i og i o S i ow T o f i o Thus the rate of change of the discrete entropy can finally be expressed as d dt vol o h o = o = using the relation (4.8). This establishes boundary nodes boundary nodes S i o(w T o f i o G i o) S i of i o (6.) Theorem 6. If the flux vector fop i is constructed so that it satisfies (6.0), the rate of change of the discrete entropy is exactly boundary nodes S i of i o where F i o is the entropy flux evaluated with the values u o. The theorem holds for any choice of the entropy function (6.6) which satisfies the convexity requirement. Equation (6.0) is satisfied exactly if the flux f i op is evaluated in the manner described in Section 4 as where f i op = 0 f i (ŵ(θ))dθ (6.) ŵ(θ) = w o + θ(w p w o ) (6.3) The entropy variables are needed only in the flux evaluation by these formulations. Aside from this equation (6.7) represents a standard finite volume approximation of (6.) in 45

46 conservation form. Because the entropy variables, such as those specified in equation (6.), contain fractional powers of p and ρ, it appears that one must resort to numerical integration to evaluate (6.). For this purpose the interval of integration may be shifted to [, ] as in Section 5, so that fop i is expressed as fop i = where f i ( w(θ))dθ w(θ) = (w p + w o ) + θ(w p w o ) Then we may use the n point Lobatto formula. The 3 point formula is Simpson s rule fop i = [f( w( )) + 4f( w(θ)) + f( w())] 6 The 5 point formula f i op = 80 [9(f( w( )) + f( w())) + 49(f( w( θ a)) + f( w(θ a ))) + 64f( w(0))] where θ a = 7 appears to be a good compromise between accuracy and computational cost. 46

47 7 Conclusion The foregoing sections demonstrate that both scalar conservation laws and systems of conservation laws which satisfy an entropy principle can be approximated in semi-discrete conservation form in a manner that also globally satisfies the corresponding discrete energy or entropy principle. This provides a path to the construction of stable non-dissipative discrete operators. However, if the governing equations support weak solutions containing shock waves (Burgers equation, gas dynamics), the energy or entropy principle is no longer valid in its basic form, and must be modified to account for energy dissipation or entropy production by the shock waves. Correspondingly the discrete formulation must be augmented by shock operators which restore the energy or entropy balance. Since the inception of finite volume methods in computational fluid dynamics [4], there has been an issue of whether it would be better to calculate the interface flux by averaging the flux vectors f j+ = (f j+ + f j ) or by calculating the flux from the average of the state vectors f j+ = f ( ) (u j+ + u j ) Theorem (4.3) suggests that neither is the best choice. Instead, the use of equation (4.4) for the evaluation of the numerical flux is consistent with the discrete entropy principle. In the course of the original development of the Jameson-Schmidt-Turkel (JST) scheme [5], which has been widely used to calculate transonic and supersonic flows, the scheme which was initially tested added artificial diffusion only in regions with strong pressure gradients, such as the neighborhood of shock waves. However, in numerous numerical experiments it appeared that the use of non-reflecting boundary conditions was not sufficient to assure convergence to a completely steady state, with the residuals reduced to machine zero. This led to the introduction of a higher-order background diffusive terms which were switched off at shock waves to prevent oscillations. It now appears worthwhile to rexamine whether satisfaction of the discrete entropy principle would allow the background dissipation to be substantially reduced or eliminated in simulations of both steady and unsteady flows. Certainly any artificial diffusive terms in a compressible large eddy simulation will need to be very carefully controlled in order to obtain correct energy spectra. 47